• Soha Othman
• El Houssainy A. Rady
• Ahmed Amin El-Sheikh

The coefficient of determination (R^2 ) and its adjusted version (R ̅^2 ) are fundamental measures of goodness of fit in regression analysis. Their exact sampling distributions have been extensively studied under the assumption of normally distributed errors. However, many real-world data exhibit heavy tails, for which the normal assumption is unrealistic. This paper derives the exact density function, cumulative distribution function (CDF), and raw moments of R^2 and R ̅^2 when the error term follows a multivariate standard Cauchy distribution. The derivation represents the Cauchy distribution as a member of the scale mixture of normals (SMN) class with an inverse-gamma mixing distribution. Numerical investigations are conducted to assess the behavior of the mean, bias, and standard error of R² and R̅² under varying sample sizes, numbers of regressors, and values of the population parameter ϕ, both with and without specification error. Results indicate that while R^2 tends to overstate fit and becomes unstable under heavy-tailed errors, adjusted R^2 provides a more robust and reliable measure of model fit.

Linear regression, coefficient of determination R^2, adjusted R ̅^2, multivariate Cauchy distribution

"The Distribution of the R^2 and (R^2 ) ̅ When the Error Term has a Multivariate Standard Cauchy Distribution", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.12, Issue 10, page no.b329-b333, October-2025, Available :http://www.jetir.org/papers/JETIR2510148.pdf

"The Distribution of the R^2 and (R^2 ) ̅ When the Error Term has a Multivariate Standard Cauchy Distribution", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.12, Issue 10, page no. ppb329-b333, October-2025, Available at : http://www.jetir.org/papers/JETIR2510148.pdf